Optimal. Leaf size=416 \[ 3 i b c^2 d^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-3 i b c^2 d^3 \text{PolyLog}\left (2,-1+\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+3 b^2 c^2 d^3 \text{PolyLog}\left (2,-1+\frac{2}{1-i c x}\right )+b^2 c^2 d^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )+\frac{3}{2} b^2 c^2 d^3 \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )-\frac{3}{2} b^2 c^2 d^3 \text{PolyLog}\left (3,-1+\frac{2}{1+i c x}\right )+\frac{7}{2} c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2-i c^3 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-2 i b c^2 d^3 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+6 i b c^2 d^3 \log \left (2-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-6 c^2 d^3 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-\frac{b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac{1}{2} b^2 c^2 d^3 \log \left (c^2 x^2+1\right )+b^2 c^2 d^3 \log (x) \]
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Rubi [A] time = 0.752747, antiderivative size = 416, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 20, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {4876, 4846, 4920, 4854, 2402, 2315, 4852, 4918, 266, 36, 29, 31, 4884, 4924, 4868, 2447, 4850, 4988, 4994, 6610} \[ 3 i b c^2 d^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-3 i b c^2 d^3 \text{PolyLog}\left (2,-1+\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+3 b^2 c^2 d^3 \text{PolyLog}\left (2,-1+\frac{2}{1-i c x}\right )+b^2 c^2 d^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )+\frac{3}{2} b^2 c^2 d^3 \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )-\frac{3}{2} b^2 c^2 d^3 \text{PolyLog}\left (3,-1+\frac{2}{1+i c x}\right )+\frac{7}{2} c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2-i c^3 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-2 i b c^2 d^3 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+6 i b c^2 d^3 \log \left (2-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-6 c^2 d^3 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-\frac{b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac{1}{2} b^2 c^2 d^3 \log \left (c^2 x^2+1\right )+b^2 c^2 d^3 \log (x) \]
Antiderivative was successfully verified.
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Rule 4876
Rule 4846
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 4852
Rule 4918
Rule 266
Rule 36
Rule 29
Rule 31
Rule 4884
Rule 4924
Rule 4868
Rule 2447
Rule 4850
Rule 4988
Rule 4994
Rule 6610
Rubi steps
\begin{align*} \int \frac{(d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x^3} \, dx &=\int \left (-i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x^3}+\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x^2}-\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}\right ) \, dx\\ &=d^3 \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x^3} \, dx+\left (3 i c d^3\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x^2} \, dx-\left (3 c^2 d^3\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{x} \, dx-\left (i c^3 d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-i c^3 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-6 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )+\left (b c d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{x^2 \left (1+c^2 x^2\right )} \, dx+\left (6 i b c^2 d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx+\left (12 b c^3 d^3\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx+\left (2 i b c^4 d^3\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=4 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-i c^3 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-6 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )+\left (b c d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{x^2} \, dx-\left (6 b c^2 d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{x (i+c x)} \, dx-\left (2 i b c^3 d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx-\left (b c^3 d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx-\left (6 b c^3 d^3\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx+\left (6 b c^3 d^3\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx\\ &=-\frac{b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac{7}{2} c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-i c^3 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-6 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )-2 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )+6 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )+3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )-3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )+\left (b^2 c^2 d^3\right ) \int \frac{1}{x \left (1+c^2 x^2\right )} \, dx+\left (2 i b^2 c^3 d^3\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (3 i b^2 c^3 d^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx+\left (3 i b^2 c^3 d^3\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (6 i b^2 c^3 d^3\right ) \int \frac{\log \left (2-\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx\\ &=-\frac{b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac{7}{2} c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-i c^3 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-6 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )-2 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )+6 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )+3 b^2 c^2 d^3 \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )+3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )-3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )+\frac{3}{2} b^2 c^2 d^3 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )-\frac{3}{2} b^2 c^2 d^3 \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )+\frac{1}{2} \left (b^2 c^2 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )+\left (2 b^2 c^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )\\ &=-\frac{b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac{7}{2} c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-i c^3 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-6 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )-2 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )+6 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )+3 b^2 c^2 d^3 \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )+b^2 c^2 d^3 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )+3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )-3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )+\frac{3}{2} b^2 c^2 d^3 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )-\frac{3}{2} b^2 c^2 d^3 \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )+\frac{1}{2} \left (b^2 c^2 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\frac{1}{2} \left (b^2 c^4 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac{b c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac{7}{2} c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{2 x^2}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-i c^3 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-6 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac{2}{1+i c x}\right )+b^2 c^2 d^3 \log (x)-2 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )-\frac{1}{2} b^2 c^2 d^3 \log \left (1+c^2 x^2\right )+6 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1-i c x}\right )+3 b^2 c^2 d^3 \text{Li}_2\left (-1+\frac{2}{1-i c x}\right )+b^2 c^2 d^3 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )+3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )-3 i b c^2 d^3 \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )+\frac{3}{2} b^2 c^2 d^3 \text{Li}_3\left (1-\frac{2}{1+i c x}\right )-\frac{3}{2} b^2 c^2 d^3 \text{Li}_3\left (-1+\frac{2}{1+i c x}\right )\\ \end{align*}
Mathematica [A] time = 1.07768, size = 500, normalized size = 1.2 \[ \frac{1}{2} d^3 \left (-6 i a b c^2 (\text{PolyLog}(2,-i c x)-\text{PolyLog}(2,i c x))-2 i b^2 c^2 \left (\tan ^{-1}(c x) \left ((c x-i) \tan ^{-1}(c x)+2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )-i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )\right )+6 b^2 c^2 \left (-i \tan ^{-1}(c x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(c x)}\right )-i \tan ^{-1}(c x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )-\frac{1}{2} \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(c x)}\right )+\frac{1}{2} \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(c x)}\right )-\frac{2}{3} i \tan ^{-1}(c x)^3-\tan ^{-1}(c x)^2 \log \left (1-e^{-2 i \tan ^{-1}(c x)}\right )+\tan ^{-1}(c x)^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+\frac{i \pi ^3}{24}\right )+\frac{6 b^2 c \left (c x \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(c x)}\right )+\tan ^{-1}(c x) \left ((c x-i) \tan ^{-1}(c x)+2 i c x \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )\right )\right )}{x}-2 i a^2 c^3 x-6 a^2 c^2 \log (x)-\frac{6 i a^2 c}{x}-\frac{a^2}{x^2}-2 i a b c^2 \left (2 c x \tan ^{-1}(c x)-\log \left (c^2 x^2+1\right )\right )-\frac{6 i a b c \left (c x \left (\log \left (c^2 x^2+1\right )-2 \log (c x)\right )+2 \tan ^{-1}(c x)\right )}{x}-\frac{2 a b \left (\tan ^{-1}(c x)+c x \left (c x \tan ^{-1}(c x)+1\right )\right )}{x^2}-\frac{b^2 \left (-2 c^2 x^2 \log \left (\frac{c x}{\sqrt{c^2 x^2+1}}\right )+\left (c^2 x^2+1\right ) \tan ^{-1}(c x)^2+2 c x \tan ^{-1}(c x)\right )}{x^2}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 2.691, size = 1846, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{-4 i \, a^{2} c^{3} d^{3} x^{3} - 12 \, a^{2} c^{2} d^{3} x^{2} + 12 i \, a^{2} c d^{3} x + 4 \, a^{2} d^{3} +{\left (i \, b^{2} c^{3} d^{3} x^{3} + 3 \, b^{2} c^{2} d^{3} x^{2} - 3 i \, b^{2} c d^{3} x - b^{2} d^{3}\right )} \log \left (-\frac{c x + i}{c x - i}\right )^{2} +{\left (4 \, a b c^{3} d^{3} x^{3} - 12 i \, a b c^{2} d^{3} x^{2} - 12 \, a b c d^{3} x + 4 i \, a b d^{3}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{4 \, x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} d^{3} \left (\int \frac{a^{2}}{x^{3}}\, dx + \int - i a^{2} c^{3}\, dx + \int - \frac{3 a^{2} c^{2}}{x}\, dx + \int \frac{b^{2} \operatorname{atan}^{2}{\left (c x \right )}}{x^{3}}\, dx + \int \frac{3 i a^{2} c}{x^{2}}\, dx + \int - i b^{2} c^{3} \operatorname{atan}^{2}{\left (c x \right )}\, dx + \int \frac{2 a b \operatorname{atan}{\left (c x \right )}}{x^{3}}\, dx + \int - \frac{3 b^{2} c^{2} \operatorname{atan}^{2}{\left (c x \right )}}{x}\, dx + \int - 2 i a b c^{3} \operatorname{atan}{\left (c x \right )}\, dx + \int \frac{3 i b^{2} c \operatorname{atan}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int - \frac{6 a b c^{2} \operatorname{atan}{\left (c x \right )}}{x}\, dx + \int \frac{6 i a b c \operatorname{atan}{\left (c x \right )}}{x^{2}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (i \, c d x + d\right )}^{3}{\left (b \arctan \left (c x\right ) + a\right )}^{2}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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